Final answer:
The maximum number of turning points for the function f(x)=-10x^6+11x^5+x^7+12 is 2.
Step-by-step explanation:
The maximum number of turning points for the function f(x)=-10x^(6)+11x^(5)+x^(7)+12 can be determined by analyzing its graph. Turning points occur when the graph changes direction from increasing to decreasing or vice versa. To find the maximum number of turning points, we need to find the number of times the graph changes direction.
To do this, we can take the derivative of the function and find the critical points. The critical points occur where the derivative is equal to zero or undefined. We can then test the intervals between the critical points and determine the direction of the graph in each interval.
In this case, the derivative of the function is f'(x)=-60x^(5)+55x^(4)+7x^(6). By setting the derivative equal to zero, we can find the critical points: x=0 and x≈0.6667. By testing the intervals between the critical points, we can determine that the graph changes direction twice, resulting in a maximum of 2 turning points.
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